Magnetic fields

By the end of this chapter you should be able to:

•understand the meaning of magnetic fied and find its magnitude and direction in simple situations involving stright-line conductors and solenoids using the right-hand rule where appropriate;•find the force on moving charges and currents in magnetic fields and appreciate the definition of the ampere as a fundamental SI unit, using the right-hand rule for forces where appropriate.

Magnetic fieldBoth magnets and electric currents create magnetic fields around themselves and when another magnet or electric current (or moving charge in general) enters this magnetic field it will experience a magnetic force. The magnetic field is a vector quantity just like the electric field.

The direction of the magnetic field

The magnetic field direction is determined by the effect it has on a compass needle. A magnetic needle aligns itself in the direction of the magnetic field vector.

Magnetic field linesJust like electric field lines, magnetic field lines are defined as imaginary lines around magnets and currents, tangents to which give the direction of the magnetic field.

Magnetic field lines of permanent magnets

Magnetic field lines created by a solenoid

Magnetic field lines

The magnetic field lines of a single turn of wire

Magnetic field lines for a straight current wire

The magnetic force on a currentIf a current is placed in a region of magnetic field, it will experience a magnetic force. The magnitude of the force is proportional to the current I, the magnetic field magnitude B and the lenth L of the wire that is in the magnetic field.

The force on a lenght L of the wire is given by

F = BIL sinθ,

where θ is the angle between the current and the direction of the magnetic field. The direction of the magnetic force is always normal to both the current and the magnetic fied.

The magnetic force on a moving charge

An electric current that is in a magnetic field will experience a force. But an electric current is just moving charges, so a moving charge will experience a magnetic force as well.

Consider a positive charge q that moves with speed v to the right. In time Δt the charge will move a distance L = v Δt. The current created by this charge is I=q/Δt, so the force on this current is

F = BIL sinθ = B(q/Δt)vΔt sinθ = qvB sinθ

This implies that the magnetic force is zero if the charge moves parallel or antiparallel to the magnetic field. There is also no magnetic force if the charge is not moving. The magnetic force on particles that are electrically neutral is zero.

Orsted’s discoveryThe magnitude of the magnetic field B created by the current in a wire varies linearly with the current in the wire and inversely with the perpendicular distance from the wire.

B = μ0 I/(2πr)

μ0 = 4π x 10-7 Tm/A(magnetic permeability of vacuum)

Orsted’s discoveryThe magnetic field strength B at the centre of a circular loop of radius R carrying current I is

In the interior of the solenoid the magnetic field is uniform in magnitude and direction and is given by

B = μ0 NI/L.

A much stronger magnetic field can be obtained if the solenoid has an iron core.

The force between two current-carrying wires

Consider two long, straight, parallel wires each carrying current, say I1 and I2. The first wire (1) creates a magnetic field in space, and in particular at the position of the second wire (2). Thus, wire 2 will experience a magnetic force. Similarly, wire 2 will produce a magnetic field a the position of wire 1, so that wire 1 will also experience a magnetic force. By Newton’s third law, the forces experienced by the two wires are equal and opposite. If the currents are parallel, the forces are attractive and if they are antiparallel, the forces are repulsive.

The force between two current-carrying wires

The wire 1 creates a magnetic field B1 and the wire 2 a magnetic field B2. This means that wire 2 in the magnetic field of wire 1 (B1), and so will experience a force. Similarly, wire 1 is in the magnetic field of wire 2 (B2), and so it too will experience a force. If the two parallel wires are separated by a distance r, then

Now, the force on a length L of wire 2 is

and similarly the force on an equal length of wire 1 is

Ampere definition

The ampere is defined through the magnetic force between two parallel wires. If the force on a 1 m length of two wires that are 1 m apart and carrying equal currents is 2 x 10-7 N, then the current in eache wire is defined to be 1 A.